Upgrading existing buildings to meet modern seismic standards is one of the greatest challenges in structural engineering today. This case study highlights how a typical 1970s reinforced concrete hotel was transformed into a safe, resilient, and code-compliant structure—without compromising its functionality or architectural value. The project involved an 8-storey, 8,400 m² hotel, where seismic strengthening was successfully integrated into a full architectural renovation. (Figure 1).

Figure 1. Front view of the building before the seismic retrofit and the architectural renovation

The building was analysed both before and after the strengthening interventions in accordance with Eurocode 8, Part 3, using the SeismoBuild software. SeismoBuild is an advanced finite element package dedicated to the linear and nonlinear assessment and strengthening of existing buildings. It employs the solution algorithms of SeismoStruct while automating the entire process—from intuitive structural modelling and linear (RSA) or nonlinear (pushover and dynamic) analysis, to code-based verification, CAD drawing export, and technical report generation. It is the ultimate tool for engineers who need to carry out code-compliant structural vulnerability assessment and seismic retrofit design of existing buildings, with speed, accuracy, and scientific rigor.

A comparison between SeismoBuild and SeismoStruct can be found here.

The buildings’ structural model without the infill walls is shown in Figure 2.

Figure 2. Structural model of the building (SeismoBuild screenshots)

The hotel exhibited a series of structural deficiencies.

The most prominent feature of the building was its high irregularity in both plan and elevation. In plan view, one horizontal dimension was significantly larger than the other. Furthermore, the vertical load-bearing system consisted of numerous relatively small and weak columns, and no substantial structural walls were present, except for a lightly reinforced concrete core wall located at the perimeter of the elevator shaft, positioned on one side of the building (Figure 3).

Figure 3. Plan view of the building’s typical floor. The columns are shown in blue and the concrete walls in orange (SeismoBuild screenshot)

In elevation, the deficiencies were even more critical. The building featured a typical soft ground storey with large open spaces and no infill walls. Additionally, directly above the soft storey, there was a short storey (h = 1.50 m) with reinforced concrete walls along its entire perimeter, originally intended to house mechanical installations. This configuration significantly accentuated vertical irregularities and amplified higher-mode effects (Figure 4). Furthermore, two columns (one at the front and one at the rear) terminated at the ground floor for architectural and operational reasons, allowing for large open spaces in the reception area. These columns were supported indirectly by two large beams at the level of the short storey. However, despite their large dimensions (150/50 cm), these beams were lightly reinforced and relatively weak (Figure 5).

Figure 4. Front view of the building, where the stiff, short second floor is shown

Figure 5. Front view of the building, where the indirect support of the columns on the large beams is shown

The column and beam dimensions were generally small, compared to modern standards, and the transverse reinforcement ratios were low with stirrups ∅8mm/250-300 and S220 steel grade. Combined with the absence of structural walls, this resulted in a highly flexible structure: the top displacement for the design earthquake was very close to the yield displacement (the design ground acceleration in the region is ag=0.24), as shown in Figure 6.

Figure 6. Typical pushover analyses (a) in the X-X direction and (b) in the Y-Y direction

Such large deformations might be acceptable in modern structures that are designed with high ductility. However, in older, low-ductility structures, this behaviour is unacceptable, indicating the necessity for structural upgrading. Moreover, a significant portion of the deformation was concentrated at the ground floor, which is a typical soft-storey behaviour. The interstorey drift at this level exceeded 1.0% under the design earthquake, signifying severe localized damage

As an initial strengthening measure, all vertical members were upgraded using reinforced concrete (RC) jacketing. A purely global strengthening strategy (e.g., adding only new RC walls or steel braces) was not considered sufficient, as the existing columns were deemed too weak to remain unstrengthened (Figure 7).

Figure 7. Strengthening of the columns with reinforced concrete jackets

In addition, new shear walls were introduced to increase lateral stiffness and eliminate plan irregularities. Along the short side of the building (Y-direction), two large shear walls, each 7.00 m wide, were constructed at both ends of the structure (Figure 8).

Figure 8. Strengthening of the new shear walls in the Y-direction

Along the long side (X-direction), however, the placement of large walls was not feasible, as this would interfere with the hotel’s operation by obstructing access to rooms. Instead, at six locations on the rear side, the jacketed columns were extended laterally to form wing walls in adjacent bays (Figure 9). Although these wing walls were smaller (2.0 m wide each), their combined contribution provided sufficient stiffness in the X-direction. All walls were extended along the full height of the building to avoid introducing vertical irregularities.

Figure 9. Strengthening of the new wing walls in the Y-direction

The existing lightly reinforced walls around the elevator shaft were strengthened using FRP (Fiber Reinforced Polymer) fabrics (Figure 10). RC jacketing was avoided in this case to prevent reducing the shaft dimensions, which were already too small for modern elevator standards.

Figure 10. FRP wrapping of the core walls in the perimeter of the elevator shaft

The two large beams of the ground floor that provided indirect support to the upper columns were strengthened with strong reinforced concrete jackets. Additionally, several other beams were strengthened using either four-sided RC jacketing or three-sided FRP wrapping (Figure 11).

Figure 11. Strengthening of the beams

The SeismoBuild model incorporating all strengthening interventions is shown in Figure 12.

Figure 12. Structural model of the strengthened building

Figure 13 presents typical capacity curves obtained from pushover analyses in both the X and Y directions. In addition to a substantial increase in lateral strength (by a factor of 2.5–3.0), there is a notable increase in stiffness and a reduction in target displacement. The target displacement now corresponds to approximately 50–60% of the maximum capacity. Combined with the enhanced ductility of the strengthened members, this indicates a significantly improved structural performance.

The building now meets modern seismic standards, offering safety, reliability, and long-term value. More importantly, the retrofit strategy has ensured significant structural upgrade without compromising the architectural layout and the operational functionality of the building.

Figure 13. Typical pushover curves in the X and the Y-direction

Typical cross-sections of the strengthened components are shown in Figure 14.

Figure 14. Typical cross sections from the design of the interventions

The design of strengthening interventions presents significant challenges for engineers, both at the design stage and during construction. The careful identification of the technical, operational, social, and performance criteria involved is essential to achieving an optimal retrofit solution.

In practice, compromises are often unavoidable, requiring balanced trade-offs between competing objectives to deliver the best overall outcome. Specialized knowledge, practical experience, and sound engineering judgment are crucial in order to objectively evaluate the advantages and limitations of alternative solutions and to select the most appropriate strategy. In many cases, a combination of different strengthening methods is necessary to achieve optimal performance, ensuring a substantial improvement in structural capacity, stiffness, and ductility.

Alfakat S.A. (2026). Strengthening of an 8-storey hotel in Chalkida, Greece. Available at:  https://www.alfakat.gr/en/project/seismic-retrofit-on-an-8-storey-hotel-in-chalkida/ (Accessed: April 13, 2026).

Antoniou, S. (2023). Seismic Retrofit of Existing Reinforced Concrete Buildings, 1st Edition. Print ISBN: 9781119987321, Online ISBN:9781119987352, DOI:10.1002/9781119987352. John Wiley & Sons Ltd.

CEN (2004). EN 1998-1:2004. Design of structures for earthquake resistance -Part 1: General rules, seismic actions and rules for buildings, Comité Européen de Normalisation, Brussels.

CEN (2005). EN 1998-3:2005. Eurocode 8: Design of structures for earthquake resistance – Part 3: Assessment and retrofitting of buildings, Comité Européen de Normalisation, Brussels.

SeismoBuild (2026). SeismoBuild – A computer program for the linear and nonlinear analysis of Reinforced Concrete and Steel Buildings.

We are pleased to invite all structural engineers, researchers, and industry professionals to our upcoming free webinar, “SeismoBuild in Action.” This session is designed to give you a comprehensive walkthrough of SeismoBuild, our premier software solution dedicated to the assessment and strengthening of existing reinforced concrete structures.

Whether you are new to the software or looking to refine your existing workflow, this webinar will demonstrate how to go from a blank slate to a fully compliant retrofit report efficiently.

Event Details

• Date: Thursday, 12 March 2026
• Time: 16:00 CET
• Duration: 90 minutes (Presentation + Live Q&A)
• Format: Online Webinar
• Language: English
• Cost: Free of Charge

What You Will Learn

In this practical, hands-on session, our experts will guide you through the complete assessment lifecycle:

• Efficient Modeling: How to quickly set up and define a structural model in SeismoBuild.
• Defining Parameters: Best practices for assigning materials, structural members, and load cases.
• Code-Compliance: Performing rigorous code-based checks and accurately interpreting the analysis results.
• Automated Deliverables: How to automatically generate detailed CAD drawings and technical descriptions for your project.
• Live Interaction: A dedicated Q&A session to answer your specific technical questions.

How to Register

Reserve your seat today to ensure you don’t miss this opportunity to enhance your expertise in seismic retrofitting.

Can’t make it live? We understand that schedules can be busy. Please register anyway, and we will send a recording of the full session to your email after the event concludes.

We look forward to welcoming you online on March 12th!

Theoretical models are only as good as their ability to predict reality. Following our exploration of the theory behind Nonlinear Analysis, this article puts those concepts to the test. Here, we present four comparative case studies where analytical predictions from SeismoStruct and SeismoBuild are benchmarked against full-scale experimental results. From the iconic 4-storey ICONS frame tested at ELSA to a 7-storey shear wall building tested at UCSD, we demonstrate the accuracy and reliability of nonlinear modeling in capturing displacement demands and failure mechanisms in practical engineering scenarios.

In the subsequent sections a series of examples is presented, whereby analytical predictions are compared against measured quantities of the true dynamic behaviour, as measured during experimental tests. In all cases, SeismoStruct was employed for the execution of the nonlinear analyses, however similar results could have been obtained with SeismoBuild, which shares the same analytical engine with SeismoStruct.

All the structural models employed feature nonlinear characteristics that enable them to correctly identify the structural behaviour and failure mechanisms in the highly-inelastic range. Material inelasticity is represented through fibre modelling with the distributed plasticity approach. Different types of frame elements were employed, e.g. force-based, force-based plastic hinge or displacement-based element types, and geometric nonlinearities are automatically incorporated in the model by the program.

This example describes the modelling of a full-scale, four-storey, 2D bare frame, which was designed essentially for gravity loads and a nominal lateral load of just 8% of its weight (Fig. 1). The reinforcement details attempted to reproduce the construction practices used in southern European countries in the 1950’s and 1960’s. The building was modelled as a plane, three-bay RC frame. The dimensions are indicated in Fig. 2.

The frame was tested at the ELSA laboratory (Joint Research Centre, Ispra) under two subsequent pseudo-dynamic loadings. Further information about the ICONS frame and the tests conducted in ELSA, can be found in Pinto et al. (1999), Carvalho et al. (1999), Pinho and Elnashai (2000) and Varum (2003).

Fig. 1: ICONS frame tested at the ELSA laboratory of Ispra

Fig. 2: Four-storey, three-bay RC frame geometry (elevation and plan views, Carvalho et al. [1999])

The analytical results, obtained with the FE analysis program SeismoStruct, are compared with the experimental results. Two different models were created:

• Model A: Both columns and beams were modelled through 3D force-based inelastic frame elements with 4 integration sections.
• Model B: Both columns and beams were modelled through 3D displacement-based inelastic frame elements.

The masses, proportional to the tributary areas, were applied either (i) as lumped masses at each beam-column joint or (ii) as distributed masses along the beams.
Two artificial records (Acc475 and Acc975, with 475 and 975 year return periods respectively) separated by a 35 sec interval, were run in series for the dynamic time-history analysis. The two input motions are given in Fig. 3 and Fig. 4. The FE model is presented in Fig. 5.

Fig. 3: Artificial acceleration time-history for 475 year return period (Acc-475)

Fig. 4: Artificial acceleration time-history for 975 year return period (Acc-975)

Fig. 5: FE model of ICONS frame in SeismoStruct

The comparison between experimental and analytical results, in terms of total displacement against time is shown in Fig. 6 and Fig. 7. The analysis provided a very good estimate of the structural response throughout the duration of the vibration for both records. It is noted that in the large Acc-975 record, the test was stopped for safety reasons, due to the heavy damage sustained by the building.

Fig. 6: Experimental vs. Analytical results – top displacement vs. time (475yrp)

Fig. 7: Experimental vs. Analytical results – top displacement vs. time (975yrp)

This example case study concerns a prototype building, seven-storey, full-scale RC shear wall frame presented in Fig. 8, tested on the NEES Large High-Performance Outdoor Shake Table at UCSD’s Englekirk Structural Engineering under dynamic conditions (Panagiotou et al. 2006), by applying four subsequent uniaxial ground motions.

Fig. 8: Seven-storey, full-scale RC shear wall building tested at the NEES Large High-Performance Outdoor Shake Table at UCSD’s Englekirk Structural Engineering Center (Martinelli P. and Filippou F.C., 2009)

As depicted in Fig. 8 and Fig. 9, the frame consists of (i) a cantilever web wall, (ii) a flange wall, (iii) a precast segmental pier and (iv) gravity columns. At each floor, the slab is simply supported by the wall and the columns.

Fig. 9: Frame geometry (floor plan view) (Martinelli P. and Filippou F.C., 2009)

Both walls were modelled through 3D force-based inelastic frame elements with 4 integration sections. The number of fibres used in section equilibrium computations was 200. The pinned gravity columns were modelled through truss elements, where the number of fibres used in section equilibrium computations was set to 200. The precast column, since it was designed in order to remain elastic, is modelled through an elastic frame element. Finally, the modelling of the slabs was realized through rigid diaphragms. The masses were computed from the values of weights given by the organizing committee and are assigned in a lumped fashion to each floor node. Finally, the base nodes are fully restrained, so that the anchorage between the structure and the shaking table is modelled.

The applied ground motion is shown in Fig. 10, and the FE model is depicted in Fig. 11:

Fig. 10: Input ground motion

Fig. 11: FE model of the tested 7-storey RC shear wall building in SeismoStruct

The comparison between experimental and analytical results is shown in Fig. 12 and Fig. 13, for both the total displacement and the base shear time-histories. Again, the analysis provided a very good estimate of the structural response for the entire record, even for large drifts and very high levels of inelasticity.

Fig. 12: Experimental vs. Analytical results – top displacement vs. time (EQ4)

Fig. 13: Experimental vs. Analytical results – base shear vs. time (EQ4)

Example 3 presents the nonlinear modelling of a full-scale, three-storey, three-dimensional RC building, which was designed for gravity loads only, according to the 1954-1995 Greek Code. The prototype building was built with the construction practice and materials used in Greece in the early 70’s (non-earthquake resistant construction). It is regular in height but highly irregular in plan (Fig. 14 and Fig. 15). Details on the structural beam member dimensions and reinforcing bars can be found in Fardis and Negro (2006). The prototype building, shown in Fig. 14, was tested at the European Laboratory for Structural Assessment (ELSA) of the Joint Research Centre of Ispra (Italy) under pseudo-dynamic conditions using the Herceg-Novi bi-directional accelerogram registered during the Montenegro 1979 earthquake.

Fig. 14: Full-scale, three-storey prototype building (Fardis & Negro, 2006)

Fig. 15: Plan view of the full-scale, three-storey prototype building (Lanese et al., 2008)

Columns and beams were modelled through 3D displacement-based inelastic frame elements with the number of section fibres set to 200. Applied masses are distributed along columns and beams, while all foundation nodes were considered as fully restrained against rotations and translations. Slabs were modelled by introducing a rigid in-plane diaphragm for each floor level.

Two input acceleration time-histories were used, H-BCR-140 for the X and H-BCR-140 for the Y direction, which are shown in Fig. 16 and Fig. 17. The building FE model is shown in Fig. 18:

Fig. 16: H-BCR140 accelerogram in the X direction

Fig. 17: H-BCR230 accelerogram in the Y direction (b)

Fig. 18: FE model of the building in SeismoStruct

Fig. 19 presents a comparison between experimental and analytical results, in terms of total displacements, which are again in good agreement.

Fig. 19: Experimental vs. Analytical results – displacement vs. time

This is an example that describes the modelling of a prototype building (full-scale, 3D steel moment resisting frame presented in Fig. 20) tested on the world’s largest three-dimensional shaking table located at Miki City, Hyogo Prefecture (Japan) under dynamic conditions, by applying a scaled version of the near-fault motion recorded in Takatori during the 1995 Kobe earthquake. The testing was carried out in the framework of the 2007 Blind Analysis Contest announced by the executive committee of the E-Defense steel building project (NRIESDP, 2007). Additional 3D shaking table tests have been performed consecutively with increasing levels of seismic motion to evaluate the effect of plastic deformation: Takatori scaled to 40% (elastic level), Takatori scaled to 60% (incipient collapse level) and Takatori in full scale (collapse level). Details on the prototype building including geometrical and material characteristics are summarized by Pavan (2008).

The analytical results obtained using a 3-D model developed in SeismoStruct after the test (post-test results), are compared with experimental results. The modelled building is schematically presented in Fig. 21.

Fig. 20: Four-storey 3D steel moment resisting frame (NRIESDP, 2007)

Fig. 21: Four-storey 3D steel frame geometry (frontal and lateral views) (Pavan, 2008)

Two different models of the modelled building were developed:

Model A: In Model A, columns and beams were modelled through 3D force-based inelastic frame elements with 4 integration sections.
Model B: In Model B, columns and beams were modelled through 3D displacement-based inelastic frame elements.

The masses attached to the model were computed from the values of weights given by the organizing committee and were assigned to the beam sections as additional mass. The base nodes were fully restrained, in order to model the anchorage between the structure and the shaking table, and the slabs were modelled as rigid in plane diaphragms for each floor. The analytical model is shown in Fig. 22.

Fig. 22: Four-storey 3D steel frame geometry (3D model) in SeismoStruct

The acceleration time-history applied during the dynamic analysis had three components in the East-West, North-South and Vertical directions. Each time-history consisted of three acceleration records in series separated by a 10 sec interval between them, as shown in Fig. 23, Fig. 24 and Fig. 25.

Fig. 23: Post-test shaking table acceleration time-history (NS component)

Fig. 24: Post-test shaking table acceleration time-history (EW component)

Fig. 25: Experimental Post-test shaking table acceleration time-histories (vertical component)

The analytical and experimental results in terms of maximum relative displacement in the two horizontal directions (North-South and East-West) are shown in Fig. 26, while the results in terms of the maximum storey shear are presented in Fig. 27. Again, in all cases the experimental results are in good agreement with the numerical predictions for both models A and B.

Fig. 26: Experimental vs. Analytical results – maximum relative displacement-floor level

Fig. 27: Experimental vs. Analytical results – maximum storey shear-floor level

This is a case study that presents the modelling of the full-scale, four-storey building, designed according to initial versions of Eurocode 8 and Eurocode 2 and tested at the ELSA laboratory (Joint Research Centre, Ispra). The building was tested with under pseudo-dynamic loading using floor time-histories derived from an accelerogram recorded in the 1976 Friuli earthquake. The building is shown in Fig. 28.

Fig. 28: Four-storey 3D infilled frame tested by Negro et al. (1996)

A 3-D model of the tested building was created in SeismoStruct and consisted of three RC frames (two infilled exterior frames and one bare interior frame) in the North-South direction and three bare RC frames in the South-West direction. The geometric characteristics of the model are presented in Fig. 29 while the building FE model is presented in Fig. 30.

Fig. 29: Four-storey 3D infilled frame geometry (frontal and plan views) (Negro et al., 1996)

Fig. 30: FE model of the tested building created in SeismoStruct

The RC columns and beams were modelled with force-based inelastic frame elements with 4 integration sections, and the sections were subdivided in 200 fibres. Multiple sections were assigned to some beam elements, in order to account for the change in reinforcement between the middle of the beams and their two edges. The infill panels were modelled through a four-node masonry panel element.

The structural mass was modelled as lumped masses, which were then automatically converted to gravity loads by SeismoStruct. The slabs were modelled as rigid diaphragms. All the base nodes were considered fully restrained against rotations and translations.

In Fig. 31 the base shear time-histories, obtained during the test and computed from the analysis, are plotted against time. It is evident that there is excellent agreement between the measured values from the test and the analytical predictions.

Fig. 31: Experimental vs. Analytical results – base shear vs. time

This example describes the modelling of the full-scale reinforced concrete bridge column presented in Fig. 32. The specimen was tested on the NEES Large High-Performance Outdoor Shake Table at UCSD’s Englekirk Structural Engineering Center under dynamic conditions, as part of a blind prediction contest. The geometric characteristics of the column are shown in Fig. 33, while further details can be found in Bianchi et al. (2011). Six uniaxial earthquake ground motions, starting with low-intensity shaking, were increased, so as to bring the pier progressively to near-collapse conditions.

SeismoStruct was employed for the specimen modelling by the winner in the Practitioners category, and was also used by two other teams that received an ‘Award of Excellence’, amongst a total of 41 entries.

Fig. 32: Full-scale reinforced concrete bridge column tested on the NEES Large High-Performance Outdoor Shake Table at UCSD’s Englekirk Structural Engineering Center

Fig. 33: Pier cross section and bridge pier specimen configuration (Bianchi et al., 2011)

The FE model of the pier consisted of a single 3D force-based inelastic frame element, and the number of fibres used in section equilibrium computations was set to 300. The self-mass of the pier along with a lumped mass of 228 ton concentrated at the top of the pier (see Fig. 33) were taken into consideration. The FE model is presented in Fig. 34.

Fig. 34: FE model of the bridge column in SeismoStruct

During the nonlinear dynamic analysis of the model, an acceleration time series consisting of six ground motion records (EQ1 to EQ6) in series separated by 10 sec intervals were generated. The time series used in the analysis are shown in Fig. 35, Fig. 36 and Fig. 37.

Fig. 35: Input ground motion (EQ1 and EQ2)

Fig. 36: Input ground motion (EQ3 and EQ4)

Fig. 37: Input ground motion (EQ5 and EQ6)

The comparison between numerical and experimental results for the top displacement is shown in Fig. 38, Fig. 39 and Fig. 40 for EQ1, EQ3 and EQ5 respectively. The plots with the comparison for the base shear are shown in Fig. 41, Fig. 42 and Fig. 43 for EQ1, EQ3 and EQ5 respectively. Numerical and experimental results are generally in good agreement, even in the highly inelastic range.

Fig. 38: Experimental vs. Analytical results – top displacement vs. time (EQ1)

Fig. 39: Experimental vs. Analytical results – top displacement vs. time (EQ3)

Fig. 40: Experimental vs. Analytical results – top displacement vs. time (EQ5)

Fig. 41: Experimental vs. Analytical results – base shear vs. time (EQ1)

Fig. 42: Experimental vs. Analytical results – base shear vs. time (EQ3)

Fig. 43: Experimental vs. Analytical results – base shear vs. time (EQ5)

Example 7 presents the modelling of two geometrically identical one-storey, three-dimensional RC frame structures (see Fig. 44), which were designed for low and high ductility levels, according to EC8 provisions. In each structure the slab does not cover the entire span, whereas nine large, additional masses are placed on top of it in a non-symmetrical configuration. A schematic plan of the aforementioned model is presented in Fig. 45. Each specimen (i.e. model A and model B, depending on the different steel reinforcement detailing) was tested under dynamic conditions on the LNEC-3D shaking table of Lisbon (Portugal) as part of a Blind Prediction Contest organized at the 15th World Conference on Earthquake Engineering (15WCEE), by applying four input ground motions of increasing intensity levels in each horizontal direction.

SeismoStruct was employed by the winning team, selected amongst a total of 38 participating entries. The numerical results obtained by analysis of the building model developed in SeismoStruct are compared with the experimental results obtained during the testing of the specimens.

Fig. 44: One-storey, three dimensional RC frame structures tested on the LNEC-3D shaking table of Lisbon (Portugal) during the 15WCEE (LNEC team, 2012)

Fig. 45: Plan view of the prototype model with the localization of the masses (LNEC team, 2012)

For the creation of the building FE model (see Fig. 46) columns and beams were modelled through 3D inelastic force-based frame elements with 3÷5 integration sections, depending on the element configuration. The number of fibres used in section equilibrium computations was set to 200.

The additional masses on top of the slab were applied in a lumped fashion, considering also the rotational inertia. The additional masses are connected to each other and to the adjacent beams through the use of elastic frame elements with the section properties of the slab.

Two time-histories were used for the nonlinear analysis of the model, one for each loading direction (NS and EW). Each time-history consists of four acceleration time-series (EQ1, EQ2, EQ3 and EQ4) separated by 10 sec intervals. The entire time-history used for loading the model along the East-West direction is shown in Fig. 47, while the time- history used for the North-South direction is shown in Fig. 48.

Fig. 46: FE model of the structure in SeismoStruct

Fig. 47: Input ground motion in the X direction (East-West component)

Fig. 48: Input ground motion in the Y direction (North-South component)

The comparison between experimental and analytical top displacement for Model A under the EQ3 record in the East-West and North-South directions is shown in Fig. 49 and Fig. 50. Equivalent results are presented in Fig. 51 and Fig. 52 for Model B. In spite of apparent differences, numerical and experimental results seem to agree well for both models.

Fig. 49: Experimental vs. Analytical results – top displacement vs. time (EQ3_REF)_Comp X

Fig. 50: Experimental vs. Analytical results – top displacement vs. time (EQ3_REF)_Comp Y

The comparison between experimental and analytical results for model B is shown hereafter:

Fig. 51: Experimental vs. Analytical results – top displacement vs. time (EQ3_REF)_Comp X

Fig. 52: Experimental vs. Analytical results – top displacement vs. time (EQ3_REF)_Comp Y

The examples presented in this post demonstrate that the gap between experimental observation and analytical prediction can be effectively bridged when nonlinear analysis is grounded in sound physical modelling and transparent numerical assumptions. Across a broad spectrum of structural systems—ranging from simple frames to highly irregular three-dimensional buildings and bridge components—the analytical simulations reproduced measured responses with a high degree of consistency, even under severe inelastic demand. This is particularly evident in large-scale shake-table experiments and blind prediction exercises, where prior calibration to test outcomes was not possible.

From a practical standpoint, the close agreement between analytical and experimental results supports the use of nonlinear dynamic analysis as a credible decision-making tool in performance-based assessment and design. When applied within a consistent modelling philosophy and validated against experimental evidence, such analyses provide engineers with confidence not only in numerical results, but also in the structural interpretations derived from them.

Ultimately, experimental testing and analytical simulation form a continuous feedback loop: experiments validate modelling strategies, while analytical tools extend experimental insight to real-world structures that cannot be tested directly. It is within this balance—between theoretical rigor, numerical robustness, and practical engineering judgement—that nonlinear analysis proves its true value in engineering practice.

• Bianchi F., Sousa R., Pinho R. 2011. Blind prediction of a full-scale RC bridge column tested under dynamic conditions. Proceedings of 3rd International Conference on Computational Methods in Structural Dynamics and Earthquake Engineering (COMPDYN 2011). Paper no. 294, Corfu, Greece.
• Carvalho E.C., Coelho E., Campos-Costa A. 1999. Preparation of the Full-Scale Tests on Reinforced Concrete Frames. Characteristics of the Test Specimens, Materials and Testing Conditions. ICONS Report, Innovative Seismic Design Concepts for New and Existing Structures. European TMR Network, LNEC.
• Fardis, M.N. and Negro P. 2006. SPEAR – Seismic performance assessment and rehabilitation of existing buildings. Proceedings of the International Workshop on the SPEAR Project, Ispra, Italy.
• Lanese I., Nascimbene R., Pavese A., Pinho R. 2008. Numerical simulations of an infilled 3D frame in support of a shaking-table testing campaign. Proceedings of the RELUIS Conference on Assessment and Mitigation of Seismic Vulnerability of Existing Reinforced Concrete Structures, Rome, Italy.
• LNEC team [2012] “Blind Test Challenge Report”, Report, LNEC (National Laboratory for Civil Engineering), Portugal.
• Martinelli P. and Filippou F.C. 2009. Simulation of the Shaking Table Test of a Seven-Storey Shear Wall Building, Earthquake Engineering and Structural Dynamics, 38, No. 5, : 587-607.
• Negro P., Pinto A.V., Verzeletti G., Magonette G.E. [1996] PsD Test on a Four-Storey R/C Building Designed According to Eurocodes, Journal of Structural Engineering – ASCE 122(11) 1409-1417.
• National Research Institute for Earth Science and Disaster Prevention 2007. Hyogo Earthquake Engineering Research Center. Blind Analysis Contest. URL: www.bosai.go.jp/hyogo/ehyogo/index.html
• Panagiotou M., Restrepo J.I. and Englekirk R.E. 2006. Experimental seismic response of a full scale reinforced concrete wall building, Proceedings of the First European Conference on Earthquake Engineering and Seismology, Geneva, Switzerland, Paper no. 201.
• Pavan A. 2008. Blind Prediction of a Full-Scale 3D Steel Frame Tested under Dynamic Conditions, MSc Dissertation, ROSE School, Pavia, Italy.
• Pinho, R. and Elnashai, A.S. 2000. Dynamic collapse testing of a full-scale four storey RC frame. ISET Journal of Earthquake Technology Paper No. 406; 37(4) : 143-164.
• Pinto A., Verzeletti G., Molina F.J., Varum H., Pinho R., Coelho E. 1999. Pseudo-Dynamic Tests on Non-Seismic Resisting RC Frames (Bare and Selective Retrofit Frames). EUR Report, Joint Research Centre, Ispra, Italy.
• Seismosoft .2026. SeismoBuild 2026 – A computer program for static and dynamic nonlinear analysis of framed structures. Available from URL: www.seismosoft.com
• Seismosoft .2026. SeismoStruct 2026 – A computer program for static and dynamic nonlinear analysis of framed structures. Available from URL: www.seismosoft.com
• Varum, H. 2003. Seismic Assessment, Strengthening and Repair of Existing Buildings, PhD Thesis, Department of Civil Engineering, University of Aveiro

In the assessment of existing structures, traditional linear elastic analysis often falls short, failing to capture the true behavior of buildings under seismic loads—especially when damage and inelasticity occur. This article serves as the first part of our deep dive into Nonlinear Structural Analysis, focusing on the fundamental theory and numerical formulation behind these advanced methods. For real-world validation of these theories, we also created Case Studies in Nonlinear Analysis that put these theoretical concepts to the test.

From the onset of structural analysis, engineers have employed linear elastic methods, implicitly assuming small deformations, limited damage to structural members, and an approximately elastic response of all structural components. Even in today’s practice, elastic methods are still widely used for the design of new structures. This is not unreasonable, considering that in new structures engineers are able to select the strength and stiffness characteristics of the structural components so as to achieve a reasonable distribution of inelasticity among different members, without large concentrations of inelastic deformations at particular, more vulnerable locations of the building.

This, together with careful detailing of the members (e.g. closely spaced stirrups in RC members or diagonal reinforcement where needed) and the adoption of a uniform behaviour factor (q-factor) that accounts for the inelastic response (implicitly assuming that inelasticity is approximately evenly distributed throughout the structure), provides an efficient and reasonably accurate framework for the design of new structures with a high level of reliability.

However, the true structural behaviour is inevitably nonlinear, and buildings designed and constructed prior to the introduction of modern seismic design codes still constitute a large percentage of the existing building stock. These structures were designed primarily for gravity loads, without specific provisions to resist seismic actions in a manner consistent with current practice. As a result, they often exhibit irregular arrangements of structural members, with uneven distributions of strength, stiffness, and mass in plan or elevation, which adversely affect their behaviour under earthquake loading (e.g. soft storeys, short columns, coupling beams between large shear walls, indirect beam supports, etc.).

Consequently, the use of elastic analysis procedures for existing buildings may lead to significant inaccuracies in estimating both the force and deformation demands of structural components. Moreover, in most cases this approximation results in an underestimation of displacement demands at locations where inelastic deformations tend to concentrate—typically the most vulnerable regions under seismic loading.

As a result, the engineering community over the last three decades has started to study the use of nonlinear structural analysis, especially under loading, which is expected to cause large inelastic deformations, such as earthquake loading.

Even today, the seismic design of buildings is predominantly based on linear elastic analysis, despite the general acknowledgment that such an approach lacks accuracy and may lead to significant underestimations of both force and deformation demands when compared with inelastic analysis. Linear analysis requires substantially less computational effort and fewer resources than nonlinear analysis. Until relatively recently (mid-1990s), the computational power available was insufficient to support the widespread adoption of nonlinear analysis in structural design and assessment. Moreover, the design methodologies currently in use were largely developed several decades ago, when nonlinear procedures were not widely applied, owing not only to limited computational resources but also to insufficient understanding of nonlinear structural behaviour within the engineering community.

Nevertheless, true structural behaviour is inherently nonlinear, characterized by a non-proportional relationship between displacements and applied loads, particularly in the presence of large deformations or material nonlinearities. This is especially relevant in seismic structural analysis, since most structures are not designed to remain elastic during maximum seismic events due to economic considerations and uncertainties in predicting seismic demand. Instead, they are expected to undergo significant inelastic deformations under strong ground motions. Consequently, structural analyses should, in principle, be treated as potentially nonlinear. The fundamental objective is to exploit structural ductility and post-elastic strength in order to satisfy prescribed performance criteria while minimizing capital investment.

Nonlinear analysis methods enable the estimation of structural response beyond the elastic range, explicitly accounting for strength and stiffness degradation associated with inelastic material behaviour and large displacements. As such, they provide a highly efficient and accurate analytical framework for capturing realistic structural behaviour, with fewer simplifying assumptions and more direct, performance-oriented design criteria.

Supported by recent advances in computing technologies and the growing availability of experimental and analytical data, the use of nonlinear structural analysis has become increasingly widespread. It now plays a central role in the assessment of existing buildings and is increasingly adopted in the design of new structures, either directly through performance-based design methodologies or indirectly through the evaluation of structures designed using advanced elastic approaches.

The first comprehensive guidelines for the application of nonlinear analysis were published in the mid-1990s, notably FEMA-273: NEHRP Guidelines for the Seismic Rehabilitation of Buildings (FEMA, 1997) and ATC-40: Seismic Evaluation and Retrofit of Concrete Buildings (ATC, 1996). Subsequent improvements were introduced in FEMA-440: Improvement of Nonlinear Static Seismic Analysis Procedures (FEMA, 2005) and FEMA-P440A: Effects of Strength and Stiffness Degradation on Seismic Response (FEMA, 2009a). Nonlinear analysis methodologies have since been incorporated into modern seismic assessment frameworks, including ASCE-41: Seismic Rehabilitation of Existing Buildings (ASCE, 2023), Eurocode 8 – Part 3 (EN 1998-3, 2004), and national codes in several countries, such as Italy, Greece, and Turkey. Furthermore, nonlinear analysis concepts have been widely employed in seismic risk assessment methodologies, most notably in HAZUS (Kircher et al., 1997a; Kircher et al., 1997b; FEMA, 2009b).

Nonlinear analysis is commonly applied in structural earthquake engineering practice in the following cases:

1. Evaluation and retrofit of existing buildings. Most existing buildings do not comply with modern seismic detailing requirements, making elastic analysis inadequate for reliable assessment and retrofit design. Nonlinear analysis enables a more realistic estimation of structural capacity and deformation demands, often allowing less conservative interventions and reduced retrofit costs. Consequently, seismic assessment of existing buildings has been a primary driver for the adoption of nonlinear analysis, particularly within performance-based assessment frameworks (Antoniou 2025a, Antoniou 2025b, Antoniou 2023)
2. Verification and performance assessment of new buildings. Nonlinear analysis is increasingly used to supplement elastic design by providing a more accurate prediction of structural response. Modern performance-based methodologies, such as ATC-58 (Applied Technology Council, 2009), employ nonlinear dynamic analysis to relate structural demands to explicit damage, loss, and performance metrics for both new and existing buildings.
3. Design of buildings with non-conventional systems or materials. The use of advanced materials and systems—such as fibre-reinforced polymers, shape-memory alloys, supplemental damping, and base isolation—often lies outside the scope of prescriptive code provisions. In such cases, nonlinear (static or dynamic) analysis is required to capture the full hysteretic behaviour and interaction of these systems.
4. Design of tall buildings in high seismicity regions. Tall buildings frequently employ nonstandard seismic-force-resisting systems and performance objectives exceeding minimum code requirements. As a result, nonlinear dynamic analysis is routinely mandated by specialized guidelines for performance-based seismic design of high-rise buildings.
5. Performance-based design with owner-defined objectives. Performance-based engineering allows owners to specify target performance levels that can be analytically evaluated and optimized through life-cycle cost considerations. Nonlinear analysis provides the accuracy and flexibility needed to reliably assess such customized performance objectives.
6. Seismic risk assessment. Nonlinear analysis is increasingly used in seismic risk and loss assessment frameworks, such as HAZUS, to develop building-specific fragility functions and improve the reliability of damage and loss predictions.

Although nonlinear analysis is undoubtedly superior in terms of the accuracy of structural response predictions, this improvement does not come without cost. The computational demands of nonlinear analysis are substantial, particularly for large-scale models subjected to dynamic loading.

Moreover, whereas linear analysis accounts primarily for the mass and stiffness distribution of structural members, nonlinear analysis requires explicit consideration of member strength, inelastic behaviour, and deformation- and force-based limit states. This necessitates the definition of component models capable of representing force–deformation relationships at both the element and system levels, including the effects of large deformations. Such modelling requires detailed knowledge of the structural configuration, as well as experienced engineers and additional effort to achieve an adequate level of modelling fidelity.

In addition, the results of nonlinear analyses can be highly sensitive to assumed input parameters and modelling choices. Without sufficient expertise in nonlinear assessment methods, engineers may draw misleading conclusions regarding structural performance. Generally, it is advisable to have clear expectations about those portions of the structure that are expected to undergo inelastic deformations, so as to use the analyses to confirm the locations of inelastic deformations.
Finally, unlike linear elastic procedures—which are well established and have been extensively validated over several decades—nonlinear inelastic analysis techniques are relatively recent and continue to evolve. Consequently, their effective application requires ongoing professional development, including continuous training, the acquisition of advanced analytical skills, and familiarity with rapidly evolving computational tools.

In linear structural analysis things are relatively simple and straightforward. Every element has linear cross-section properties EA, EI2, EI3, GJ and constant stiffness. The stiffnesses of all the elements are summed up to form the global stiffness matrix [K] of the building, which is also constant, and [K] is inverted once for the solution of the well-known equation P=[K]·u and the calculation of the nodal displacements for the different load cases. What is more, because the stiffness matrix is always positive-definite and the member strengths are considered unlimited, there will always be a mathematical solution to the problem.

Because in nonlinear analysis the structural stiffness matrix is no longer constant, but rather it is updated at every step, and because there is a strength limit in all or most of the structural members, the solution of the nonlinear equations becomes a much more complicated task. In the next sections the main points of the theoretical background of nonlinear analysis will be presented briefly.

Despite the maturity of the finite element (FE) method, the seismic assessment of buildings is performed primarily with linear finite elements (e.g. beams or rods), while two- and three-dimensional FE are very rarely utilized. Further, apart from modelling the frame members with beam-column elements, the numerical model of a structure should also be able to adequately capture the response of other structural components, such as infill panels, that may considerably affect the overall capacity.

The modelling of the mechanical properties of the structural members is a complex and wide-ranging subject. In linear analysis, it is sufficient to assume that the material remains linear and elastic, i.e. that the deformation process is fully reversible and the stress is a unique function of strain. However, such a simplified assumption is appropriate only within a limited range, and is gradually being replaced by more realistic approaches.

The primary source of nonlinearities in low and medium-rise building structures is material inelasticity and plastic yielding in the locations of damage. In larger high-rise buildings, while material inelasticity still plays an important role, large deformations relative to the frame element’s chord (known as P-Delta effects), and geometric nonlinearities become equally important and should be taken into account.

Material nonlinearities occur when the stress-strain or force-displacement law is not linear, or when material properties change with the applied loads. Contrary to linear analysis procedures, where the material stresses are always proportional to the corresponding strains and a fully elastic behaviour is assumed, in nonlinear analysis the material behaviour depends on current deformation state and possibly past history of the deformation. In order to estimate the stress from the strain of a particular location of the structure, complete expressions for the uniaxial stress-strain relationship of the material should be provided, including hysteretic rules for unloading and reloading.

The source of such inelasticity is defined at the sectional level, through the creation of a fibre model for the section. A fibre section consists in the subdivision of the area in n smaller areas, each of which is attributed a material stress-strain relationship, i.e. reinforcing steel and concrete, confined and unconfined. After defining the material laws of every material of the section, and calculating the stresses at the fibres, the sectional moment-curvature state of beam-column elements is then obtained through the integration of the nonlinear uniaxial stress-strain response of the individual fibres. The discretisation of a typical reinforced concrete cross-section is depicted in Fig. 1.

Estimating the inelastic response of the structural member requires the integration of the stresses calculated at appropriately selected integration cross-sections along the member (called Gauss Sections a and b, in Fig. 1). Finally, the global nonlinearity of the frame is then obtained by the assembly of the contributions in stiffness and strength of the structural components.

Fig 1: Discretisation of a typical reinforced concrete cross-section

Geometric nonlinearities involve nonlinearities in kinematic quantities, and occur due to large displacements, large rotations and large independent deformations relative to the frame element’s chord (also known as P-Delta effects).

The effect of geometric nonlinearities on the response of structures can range from negligible, in cases where large deformations are not expected, to extreme, in large and slender structures. In the general case, geometric nonlinearities must be modelled as they can ultimately lead to loss of lateral resistance, ratcheting (a gradual build-up of residual deformations under cyclic loading), and dynamic instability. Large lateral deflections magnify the internal force and moment demands, causing a decrease in the effective lateral stiffness. With the increase of internal forces, a smaller proportion of the structure’s capacity remains available to sustain lateral loads, leading to a reduction in the effective lateral strength.

For the numerical simulation of geometric nonlinearities and the inclusion of its effects in the analysis, the most advanced formulation is a total co-rotational formulation [Correia and Virtuoso, 2006], which is based on an exact description of the kinematic transformations associated with large displacements and three-dimensional rotations of the beam-column member. This leads to the correct definition of the element’s independent deformations and forces, as well as to the natural definition of the effects of geometrical non-linearities on the stiffness matrix.

In the local chord system of the beam-column element, six basic displacement degrees-of-freedom (θ2(A), θ3(A), θ2(B), θ3(B), Δ, θT) and corresponding element internal forces (M2(A), M3(A), M2(B), M3(B), F, MT) are defined, as shown in Fig. 2.

Fig. 2: Local chord system of a beam-column element

Once the governing equations of geometrically nonlinear structural analysis and the discretization of those equations by finite element methods is completed, a procedure is required for the solution of these equations. Nonlinear problems in mechanics are solved with incremental algorithms through a process that is considerably more complex than that used in common linear elastic analysis solvers.

All solution procedures of practical importance are strongly rooted in the idea of gradually advancing the solution by continuation, that is to follow the equilibrium response of the structure as the control and state parameters vary by small amounts. Various algorithms exist for handling such problems, but a common feature is that continuation is a multilevel process that involves a hierarchical breakdown into incremental steps, and iterative steps. The level of incrementation is always present in nonlinear solution procedures. To advance the solution, the entire loading stage is broken down into incremental steps, also called increments or steps. The incremental solution methods are then divided into two broad categories: (1) Purely incremental methods, also called predictor-only methods, and (2) Corrective methods, also called predictor-corrector or incremental-iterative methods.

In purely incremental methods the iteration level is missing. In corrective methods a predictor step is followed by one or more iteration steps. The set of iterations is called the corrective phase. Its purpose is to eliminate or reduce the so called drifting error, which is a serious problem of purely incremental methods. For this reason the corrective methods have become the standard for the solution of the nonlinear equations in all modern finite element packages.
Solutions accepted after each increment following a corrective phase are often of interest to users because they represent approximations to equilibrium states until the final loading state. They are therefore saved as they are computed. On the other hand, intermediate results of the iterative process are rarely of interest, since the solutions are not equilibrated and constitute an intermediate step until the next equilibrated solution. Hence, most programs discard them.

The use of increments may seem at first sight unnecessary if one is interested primarily in the final solution. But breaking up a stage into increments may serve different purposes: (i) The presence of path-dependent effects in nonlinear analysis problems severely restricts increment sizes because of history-tracing constraints. For example, in plasticity analysis stress states must not be allowed to stray too far outside the yield surface. (ii) The engineer can acquire a better insight into structural behaviour by studying the response plot toward the final solution, which in many cases can provide more useful information than simply the structural state at the end. It is noted that in several cases failures and critical points occur before the stage end. (iii) The breakdown of the entire loading stage can lead to more stable solutions and avoid convergence problems.

The basic method for the solution of nonlinear equations in the majority of finite element programs is the load-control Newton-Raphson (NR) algorithm and variations of it. The Newton-Raphson method, in its simplest form, is a numerical method for finding the roots of a function f(x). Since the method is iterative, a trial guess is made at x=xn. Evaluating the function at xn, we find that f (xn)≠0, i.e. it is not a root. If f’(xn) is the tangent of the function at xn, the equation of the tangent passing through xn, is:

With the aid of equation (1), we can obtain a second trial solution at xn+1:

where xn+1 is the point where the tangent intersects axis X. If f (xn+1) ≈ 0, then we have located the root, otherwise we proceed finding a new trial solution at xn+2, until convergence to the correct solution has been reached within an acceptable convergence limit. The method is schematically shown in Fig. 3.

Fig. 3: Application of the Newton-Raphson (NR) method for finding the roots of a function f(x)

In structural mechanics the Newton-Raphson (NR) method is extended, so that to accommodate the solution of a system of nonlinear equations, as described by the general equation of equilibrium:

Because of the nonlinear nature of the problem, the stiffness matrix K is a function of the deformation vector u and is constantly updated at each iteration, and hence the system of Eq. (3) cannot be solved directly, and should follow an incremental-iterative solution procedure that is called the Newton-Raphson method.

Fig. 4: The Newton-Raphson (NR) method in nonlinear structural analysis

The incremental-iterative Newton-Raphson method is schematically shown in Fig. 4. The iterative procedure follows the conventional scheme, whereby the internal forces corresponding to a displacement increment are computed and convergence is checked. If no convergence is achieved, then the out-of-balance forces (difference between applied load vector and equilibrated internal forces) are applied to the structure, and the new displacement increment is computed. Such loop proceeds until convergence has been achieved or the maximum number of iterations has been reached. Load-displacement plots, such as those of Fig. 4 are exact for one SDOF systems. For larger MDOF systems, they describe only schematically the structural response and the gradual convergence to the solution of the system of equations.

The full Newton-Raphson method provides a quadratic rate of convergence, meaning that it requires a small number of iterations to reach the solution. However, recalculating and inverting the stiffness matrix at every iteration requires increased computing resources. Hence, a common alternative is to recalculate and invert the stiffness matrix only at the first iteration and use it for all the corrective iterations. This approach is known as modified Newton-Raphson and is shown in Fig. 5.

The employment of Newton-Raphson (NR), modified Newton-Raphson (mNR) or NR-mNR hybrid solution procedures may lead to fairly flexible solution algorithm. It is clear that the computational savings in the formation, assembly and reduction of the stiffness matrix during the iterative process can be significant when using the mNR instead of the NR procedures. However, more iterations are often required with the mNR, thus leading, in some cases, to excessive computational effort. For this reason, the hybrid approach, whereby the stiffness matrix is updated only in the first few iterations of a load increment, does usually lead to an optimum scenario.

For further discussion and clarifications on the algorithms described above, readers are strongly advised to refer to available literature, such as the work by Cook et al. [1988], Crisfield [1991], Zienkiewicz and Taylor [1991], Bathe [1996] and Felippa [2002], to name but a few.

Fig. 5: The Modified Newton-Raphson (mNR) method in nonlinear structural analysis

In iterative-incremental solution algorithms, the iterative process at each step is continued, until ‘convergence is achieved’, that is the value of a norm of the out-of-balanced forces or the unbalanced deformations of the structure become smaller than given convergence criteria that have been specified by the user at the beginning of the analysis.

There are two distinct categories of convergence criteria in nonlinear analysis:

1. displacement/rotation-based criteria
2. force/moment-based criteria.

Two additional convergence check schemes may arise from the combination of the distinct criteria above:

3. Displacement/Rotation AND Force/Moment based scheme, where it is considered that the solution has been reached, when both the deformation and the force based criteria have been achieved
4. Displacement/Rotation OR Force/Moment based scheme, where convergence is achieved, when either the deformation or the force based criteria has been achieved.

Usually, the displacement/rotation criterion consists in verifying, for each individual degree-of-freedom of the structure, that the current iterative displacement/rotation is less or equal than a user-specified tolerance. In other words, if and when all values of displacement or rotation that result from the application of the iterative (out-of-balance) load vector are less or equal to the pre-defined displacement/rotation tolerance factors, then the solution is deemed as having converged. This concept can be mathematically expressed in the following manner:

where,

• δdi is the iterative displacement at translational degree of freedom i
• δθj is the iterative rotation at rotational degree of freedom j
• nd is the number of translational degrees of freedom
• nθ is the number of rotational degrees of freedom
• dtol is the displacement tolerance, in the employed by the analysis length unit
• θtol is the rotation tolerance, in rad (dimensionless)

The force/moment criterion, on the other hand, comprises the calculation of the Euclidean norm of the iterative out-of-balance load vector, and subsequent comparison to a user-defined tolerance factor. It is therefore a global convergence check (convergence is not checked for every individual degree-of-freedom as is done for the displacement/rotation case) that provides an indication of the overall state of convergence of the solution, and which can be mathematically described in the following manner:

where,

• Gnorm is the Euclidean norm of iterative out-of-balance load vector
• Gi is the iterative out-of-balance load at degree-of-freedom i
• VREF is the reference ‘tolerance’ value for forces (translational DOFs) and moments (rotational DOFs). Typically, different values are assumed for VREF
• for the translational and the rotational DOFs
• n is the number of the degrees-of-freedom

It is noted that, contrary to linear analysis, where a solution, even if it is unreasonable and unrealistic, can always be found, in the case of nonlinear analysis, the achievement of convergence and the calculation of the solution is not always guaranteed. This is due to purely ‘technical’ reasons (e.g. a beam cannot withstand the applied vertical loads, fails and a solution cannot be found), but also numerical reasons (e.g. if the solution procedure cannot accommodate large load increments, or sudden redistribution of forces, after the failure of a load-carrying member).

For this reason, in nonlinear analysis the maximum number of iterations is always specified, so that to avoid searching for a solution infinitely. If the maximum number of iterations is reached and convergence has not been achieved, the analysis is either stopped, or in more advanced software packages the load step is subdivided in smaller increments, in order to achieve better convergence conditions.

In addition to the convergence verification scheme described above, at the end of an iterative step three other solution checks may be carried out; numerical instability, solution divergence and iteration prediction. These criteria, all of a force/moment nature, serve the purpose of avoiding the computation of useless equilibrium iterations in cases where it is apparent that convergence will not be reached, thus minimising the duration of the analysis.

Numerical instability: The possibility of the solution becoming numerically unstable is checked at every iteration by comparing the Euclidean norm of out-of-balance loads, Gnorm, with a pre-defined maximum tolerance several orders of magnitude larger than the applied load vector (e.g. 1.0E+20). If Gnorm exceeds this tolerance, then the solution is assumed as being numerically unstable and iterations within the current increment are interrupted.
Solution divergence: Divergence of the solution is checked by comparing the value of Gnorm obtained in the current iteration with that obtained in the previous one. If Gnorm has increased, then it is assumed that the solution is diverging from a possible solution and iterations within the current increment are interrupted.

Iteration prediction: Finally, a logarithmic convergence rate check is also carried out, so as to try to predict the number of iterations required for convergence to be achieved. If this calculated number of iterations is larger than the maximum number of iterations specified by the user, then it is assumed that the solution will not achieve convergence and iterations within the current increment are interrupted.

Within the context of all modern Codes, two types of nonlinear analysis procedures are proposed for structural assessment: (i) the Nonlinear Static Procedure (NSP), also is called pushover analysis. (ii) ) the Nonlinear Dynamic Procedure (NDP), that is, the nonlinear dynamic time-history analysis.

According to ASCE 41, Section 7.4.3.1 (ASCE, 2023), in the Nonlinear Static Procedure, a mathematical model directly incorporating the nonlinear load-deformation characteristics of individual components of the building shall be subjected to monotonically increasing lateral loads representing inertia forces in an earthquake, until a target displacement is exceeded. The main objective of the method is to assess the capacity of the structure, considering both the deformability and strength of all structural members.

The lateral loads are gradually applied until the displacement of a selected ‘Control Node’, typically located at the centre of mass of the top storey of the building, reaches the so-called ‘Target Displacement’, which represents an approximation of the displacement demand under earthquake ground motion. The demand parameters for the structural components at the target displacement are then compared against the respective acceptance criteria for the desired performance state. System level demand parameters, such as story drifts and base shear forces, may also be checked.

Although the nonlinear static procedure is generally a much more reliable approach for characterizing the performance of a structure than linear procedures, it is still not exact and cannot accurately account for changes in the dynamic response as the structure degrades in stiffness; nor can it account for higher mode effects in multi-degree-of-freedom (MDOF) systems. Hence, the NSP is applicable to low-rise, regular buildings, where the response is dominated by the fundamental sway mode of vibration. It is less suitable, however, for taller, slender, or irregular buildings, where multiple vibration modes affect the behaviour.

According to ASCE 41-23, Section 7.4.4.1 (ASCE, 2023), in the Nonlinear Dynamic Procedure, a mathematical model directly incorporating the nonlinear load-deformation characteristics of individual components of the building shall be subjected to earthquake shaking represented by ground motion acceleration histories to obtain forces and displacements. The objective of the method is to assess the capacity of the structure, considering both the deformability, the strength and the hysteretic behaviour of all structural members that are subjected to the specified earthquake ground motion.

The direct integration of the equations of motion is accomplished using certain integration algorithms, such as the numerically dissipative α-integration algorithm (Hilber et al., 1977) or a special case of the former, the well-known Newmark scheme (Newmark, 1959). The modelling of seismic action is achieved by introducing acceleration loading curves (accelerograms) at the supports. In addition, dynamic analysis may also be employed for modelling of pulse loading cases (e.g. blast, impact, etc.), in which case instead of acceleration time-histories at the supports, force pulse functions of any given shape (rectangular, triangular, parabolic, and so on), can be employed to describe the transient loading applied to the appropriate nodes.

The NDP constitutes a sophisticated approach to examining the inelastic demands produced on a structure by a specific suite of ground motion acceleration histories. As with the NSP, the results of the NDP can be directly compared with test data on the behaviour of representative structural components to identify the structure’s probable performance, when subjected to a specific ground motion.

As nonlinear dynamic analysis involves fewer assumptions than the nonlinear static procedure, it is subject to fewer limitations than nonlinear static procedure. It automatically accounts for higher-mode effects and shifts in inertial load patterns as structural softening occurs. In addition, for a given earthquake record, this approach directly solves for the maximum global displacement demand produced by the earthquake on the structure, eliminating the need to estimate this demand based on general relationships.

Despite these advantages, the NDP requires considerable judgment and experience to perform, and should only be used when the engineer is thoroughly familiar with nonlinear dynamic analysis techniques and limitations. The analyses can be highly sensitive to small changes in assumptions with regard to either the character of the ground motion record used in the analysis or the nonlinear stiffness behaviour of the elements. For instance, two ground motion records enveloped by the same response spectrum can produce radically different results with regard to the distribution and amount of inelasticity predicted in the structure. Further, due to the inherent variability in earthquake ground motions, dynamic analyses for multiple ground motions are necessary to calculate an upper bound for the values of the demand parameters for a given earthquake scenario.

By more realistically representing the underlying structural mechanics, nonlinear static analysis and especially nonlinear dynamic analysis, reduce uncertainty in demand predictions, as compared to the linear methods of analysis. However, even with nonlinear dynamic analyses, it is practically impossible to always calculate accurately the demand parameters for the different structural members. Hence, very often there are discrepancies between the analytical predictions of the response parameters and their actual values during a seismic event. These discrepancies are usually largest for structural deformation and acceleration controlled quantities, and lower in force-controlled components, where internal forces are bounded by the strength of yielding members.

Since no numerical representation of the components response is exact, one of the sources of inaccuracy is the limitations related to the analytical capabilities of the selected software package itself, with which the analysis is to be carried out. Obviously, any analytical formulation that describes the hysteretic behaviour of materials, sections or members has limitations that restrain its ability to represent the structural response in a very precise manner, especially in the highly inelastic range, where the lateral stiffness is significantly reduced and the response is very sensitive to small changes of the loading. Analysts should be well aware of these limitations prior to the execution of the analyses, so that to avoid modelling strategies that magnify the possible errors and affect considerably the analytical predictions.

Nonetheless, continued advances in computational tools and modelling techniques have progressively reduced software-related inaccuracies, especially for ordinary structures with components exhibiting predictable behaviour that can be represented with reasonable accuracy. In such cases, the dominant sources of uncertainty are typically human- and data-related, namely: (i) variability and uncertainty in measured physical properties such as material characteristics, geometry, and detailing, and (ii) incomplete or imperfect representation of the actual structural behaviour in the analytical model. When combined with seismic hazard uncertainties—such as variability in ground motion intensity, frequency content, and duration—these factors can lead to substantial divergence between analytical predictions and true structural response during strong earthquakes.

In light of the above, analysts should explicitly recognize and account for the various sources of uncertainty inherent in structural evaluation. Where possible, these uncertainties should be quantified and reflected in acceptance criteria through the appropriate selection of safety or modification factors. The chosen values should be consistent with the level of confidence in both the structural information and the analytical model. For example, assessment frameworks assign different values to the knowledge factor (ASCE 41-23) or the confidence factor (Eurocode 8, Part 3), with more conservative values adopted when the available information on the structure is limited.

One means of assessing the accuracy achievable by analytical procedures is through blind prediction exercises. In such studies, engineers are asked to predict the response of a structure—whose true behaviour will be determined through experimental testing—without prior knowledge of the measured results, using state-of-the-art analytical tools. This approach provides an unbiased evaluation of predictive capability, as prior knowledge of the response cannot be used to calibrate or refine the simulations.

In recent years, numerous blind prediction exercises have been conducted, ranging from single structural components and subassemblies to large-scale, full structural systems. In many cases, teams of analysts were invited to predict the response of structures tested on shake tables, based solely on information regarding structural design and the characteristics of the imposed ground motions.

Results from these exercises have consistently shown significant scatter among predicted responses, often leading to poor agreement with measured behaviour, including nonlinear response mechanisms and failure modes. This indicates that many predictions remain unreliable in practice. Moreover, drawing definitive conclusions from such studies is challenging, as discrepancies may arise from multiple sources, including inconsistent characterization of input parameters, inaccuracies in material modelling, insufficient mesh refinement, or limitations of the adopted nonlinear modelling approaches.

Nevertheless, blind prediction exercises are invaluable for identifying and evaluating the sources of variability between analytical predictions and experimental observations, as well as for distinguishing analytical procedures that perform consistently better—or worse—than others. Careful examination of past blind prediction studies can therefore provide critical insight into the strengths and limitations of nonlinear analysis methods and contribute to the systematic improvement of their reliability.

Theoretical formulations and numerical models are only as valuable as their ability to predict real structural behavior. Now that we have established how the nonlinear methods work, the critical question remains: Do they match reality?

In the second part of this series, we put these theories to the test. We benchmark the analytical predictions of SeismoStruct and SeismoBuild against full-scale experimental results, including the famous ICONS frame tested at ELSA and a 7-storey shear wall building tested at UCSD.

Continue reading: Nonlinear Structural Analysis in Engineering Practice – Case Studies

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• [FEMA] Federal Emergency Management Agency. 1997. NEHRP Guidelines for the Seismic Rehabilitation of Buildings, FEMA 273 Report, prepared by the Applied Technology Council and the Building Seismic Safety Council for the Federal Emergency Management Agency, Washington, D.C.
• [FEMA] Federal Emergency Management Agency. 2005. Improvement of Nonlinear Static Seismic Analysis Procedures, FEMA 440 Report, prepared by the Applied Technology Council for the Federal Emergency Management Agency, Washington, D.C.
• [FEMA] Federal Emergency Management Agency. 2009a. Effects of Strength and Stiffness Degradation on Seismic Response, FEMA P-440A Report, prepared by the Applied Technology Council for the Federal Emergency Management Agency, Washington, D.C.
• [FEMA] Federal Emergency Management Agency. 2009b. HAZUS®-MH MR5 Advanced Engineering Building Module (AEBM) Technical and User’s Manual, prepared by the National Institute of Buildings Sciences for the Federal Emergency Management Agency, Washington, D.C.
• Hilber H.M., Hughes T.J.R., Taylor R.L. 1977. Improved numerical dissipation for time integration algorithms in structural dynamics. Earthquake Engineering and Structural Dynamics 5( 3) : 283-292.
• Giberson, M.F. 1967. The Response of Nonlinear Multi-Story Structures subjected to Earthquake Excitation, Doctoral Dissertation, California Institute of Technology, Pasadena, CA., May 1967 : 232
• Kircher, C. A., Nassar, A. A., Kustu, O., and Holmes, W. T. 1997a. Development of Building Damage Functions for Earthquake Loss Estimation. Earthquake Spectra 13 (4), : 663-682.
• Kircher, C. A., Reitherman, R. K., Whitman, R. V., and Arnold, C. 1997b. Estimation of Earthquake Losses to Buildings. Earthquake Spectra 13 (4) : 703-720.
• Mari A., Scordelis A. 1984. Nonlinear geometric material and time dependent analysis of three dimensional reinforced and prestressed concrete frames, SESM Report 82-12. Department of Civil Engineering, University of California, Berkeley.
• Neuenhofer A., Filippou F.C. 1997. Evaluation of nonlinear frame finite-element models. Journal of Structural Engineering 123(7) : 958-966.
• Newmark N.M. 1959. A method of computation for structural dynamics. Journal of the Engineering Mechanics Division, ASCE 85( EM3) : 67-94.
• [NIST] National Institute of Standards and Technology. 2010. Nonlinear Structural Analysis for Seismic Design, A Guide for Practicing Engineers, GCR 10-917-5, prepared by the NEHRP Consultants Joint Venture, a partnership of the Applied Technology Council and the Consortium of Universities for Research in Earthquake Engineering, for the National Institute of Standards and Technology, Gaithersburg, Maryland.
• [PEER] Pacific Earthquake Engineering Research Centre. 2010. Tall Buildings Initiative: Guidelines for Performance-Based Seismic Design of Tall Buildings, PEER Report 2010/05, Pacific Earthquake Engineering Research Center, Berkeley, California.
• PEER/ATC .2010. Modeling and acceptance criteria for seismic design and analysis of tall buildings, PEER/ATC 72-1 Report, Applied Technology Council, Redwood City, CA, October 2010.
• Willford, M., Whittaker, A., and Klemencic, R. 2008. Recommendations for the seismic design of high-rise buildings, Council on Tall Buildings and Urban Habitat, Illinois Institute of Technology, Chicago, IL.
• Seismosoft .2026. SeismoBuild 2026 – A computer program for static and dynamic nonlinear analysis of framed structures. Available from URL: www.seismosoft.com
• Seismosoft .2026. SeismoStruct 2026 – A computer program for static and dynamic nonlinear analysis of framed structures. Available from URL: www.seismosoft.com
• Scott M.H., Fenves G.L. 2006. Plastic hinge integration methods for force-based beam–column elements. ASCE Journal of Structural Engineering 132( 2) : 244-252.
• Spacone E., Ciampi V., Filippou F.C. 1996. Mixed formulation of nonlinear beam finite element. Computers & Structures 58( 1) : 71-83.
• Zienkiewicz O.C., Taylor R.L. 1991. The Finite Element Method, 4th Edition, McGraw Hill.

We are pleased to invite all structural engineers, researchers, and industry professionals to our upcoming free webinar, “SeismoBuild in Action.” This session is designed to give you a comprehensive walkthrough of SeismoBuild, our premier software solution dedicated to the assessment and strengthening of existing reinforced concrete structures.

Whether you are new to the software or looking to refine your existing workflow, this webinar will demonstrate how to go from a blank slate to a fully compliant retrofit report efficiently.

Event Details

• Date: Thursday, 5 February 2026
• Time: 16:00 CET
• Duration: 90 minutes (Presentation + Live Q&A)
• Format: Online Webinar
• Language: English
• Cost: Free of Charge

What You Will Learn

In this practical, hands-on session, our experts will guide you through the complete assessment lifecycle:

• Efficient Modeling: How to quickly set up and define a structural model in SeismoBuild.
• Defining Parameters: Best practices for assigning materials, structural members, and load cases.
• Code-Compliance: Performing rigorous code-based checks and accurately interpreting the analysis results.
• Automated Deliverables: How to automatically generate detailed CAD drawings and technical descriptions for your project.
• Live Interaction: A dedicated Q&A session to answer your specific technical questions.

How to Register

Reserve your seat today to ensure you don’t miss this opportunity to enhance your expertise in seismic retrofitting.

Can’t make it live? We understand that schedules can be busy. Please register anyway, and we will send a recording of the full session to your email after the event concludes.

We look forward to welcoming you online on February 5th!

In the context of seismic assessment and retrofit of existing reinforced concrete buildings, practitioners prefer to employ linear analysis and in particular the Linear Dynamic Procedure that uses the well-known Response Spectrum Analysis. This is attributed mainly to the fact that engineers are more familiar with this method, which is the one mostly employed in the design of new structures, but also because linear analysis is easier and faster to use.

In the design of new structures, the approximations of elastic analysis do not constitute a significant problem, since engineers are able to choose the strength and stiffness characteristics of the structural components, in order to have a reasonable distribution of inelasticity, without large concentrations of deformations at particular, more vulnerable locations of the building.

On the contrary, this is rarely the case in the context of the seismic assessment and retrofit of existing structures. Older buildings have been designed and constructed before the introduction of the early earthquake resistance codes, without special considerations to withstand seismic actions in a manner similar to today’s practice. As a result, very frequently they exhibit irregular arrangement of their structural members, with uneven distribution of the strength, stiffness and mass, which adversely affects its behavior under earthquake loading (e.g. irregularities in plan or elevation, soft ground stories, short columns, coupling beams between large shear walls, indirect supports on beams etc.).

Because of this behavior, the use of elastic procedures for the analysis of existing buildings may lead to serious inaccuracies in the estimation of the force and the deformation demand on the structural components, especially in the locations with concentrations of inelastic deformations that are the most vulnerable under seismic loading. In order to overcome all these problems, all the standards for structural assessment have proposed larger safety factors for the linear methods and procedures that are very conservative.

In the present article, it will be demonstrated that, despite the obvious advantages of linear analytical methods, their use in the assessment and strengthening of older RC structures is a very conservative practice that should be avoided. By means of examples and real case-studies that are analyzed with the full code-based seismic assessment methodologies, according to both ASCE 41 and Eurocode 8, it will be explained why the nonlinear methods, combined with a good knowledge of the structural configuration, can be very beneficial, lead to lighter interventions and prevent unnecessarily disruptive and costly works. Similar conclusions have also been drawn by other recent studies [1, 2, 3, 4].

Within the context of all modern codes for structural assessment and retrofit, four different analytical methods are proposed with small variations between the different standards:

• The Linear Static Procedure LSP, a static type of analysis with no variable load.
• The Linear Dynamic Procedure LDP, which is essentially the Response Spectrum Method (RSA), which is the method mostly employed for the design of new structures; thus, it is the method of analysis that engineers are more familiar with.
• The Nonlinear Static Procedure NSP, which is the well-know pushover analysis.
• The Nonlinear Dynamic Procedure NDP, which is the nonlinear dynamic time-history analysis.

Linear Static Procedure: The LSP is the most basic method of the four with many approximations and very limited accuracy, even for relatively simple structural configurations. It is only allowed for small symmetric buildings, and it is employed in a conservative manner with large safety factors. In general, it should be avoided for everyday application, but for the most simple and regular buildings.

Linear Dynamic Procedure: Because the loading is calculated through the combinations of several modes (including higher ones), the LDP is suitable for tall and asymmetric buildings, where higher mode effects are of importance. However, as an elastic method, it inherently has limited accuracy in the case of large inelastic deformations, which are very common in existing buildings under large earthquake loading. Hence, the results can be very inaccurate when applied to buildings with highly irregular structural systems, unless the building is capable of responding almost elastically at the selected Seismic Hazard Level. Similarly to the LSP, it is employed conservatively with higher safety margins, in comparison with the nonlinear methods.

Nonlinear Static Procedure: Due to the explicit modelling of inelasticity, the NSP is more suitable when large inelastic deformations are expected. In such cases, the structural response can be modelled with satisfactory accuracy, allowing for a less conservative approach. The NSP is generally a more reliable approach for characterizing the performance of a structure than the linear procedures. However, it cannot accurately account for changes in dynamic response as the structure degrades, and it is not suitable, when higher-mode effects are of importance, e.g. with taller buildings (more than 10-15 floors). In general, the NSP is a valid approach for the seismic assessment of existing buildings; however, it should be used with some caution, when the structural response is determined by more than one modes.

Nonlinear Dynamic Procedure: The nonlinear dynamic time-history analysis involves fewer assumptions than the nonlinear static procedure, and it is the most sophisticated method for structural analysis. It is more accurate than the NSP, and it is subject to fewer limitations regarding the load and the structural configuration. The NDP is able to model both the inelastic material behavior and higher mode effects for a given earthquake record. It directly provides the maximum global displacement demand produced by the earthquake on the structure, eliminating the need for approximations, and it is generally suitable for any structural configuration and any earthquake loading. However, the main disadvantage of the method is a significant one: it is relatively difficult to use, and specialized knowledge is often required, e.g. for the selection of suitable accelerograms, or the interpretation of results.

As in the standard design methodologies, the requirement of the capacity checks is that the component strength is larger than the demand on the component, i.e.:

QC ≥ QU for ASCE 41, or Rd ≥ Ed for the Eurocodes

QU or Ed is the design value of the action effect for the seismic design situation for the selected hazard level, and QC or Rd is the corresponding resistance of the element.

A basic distinction is done in both standards between the deformation-controlled, ductile actions (e.g. bending in a member without significant axial loads) and the force-controlled, brittle actions (e.g. shear). Different approaches are followed for the capacity checks in the two cases, as will be described below.

When using the linear methods of analysis, the checks are always performed in terms of forces for both the deformation and the force-controlled actions. For the ductile mechanisms of failure, inelasticity is taken into account in the two codes in a similar, but not exactly the same way. In ASCE 41, inelasticity is taken into account by the so-called m-factors, whereas in Eurocode 8 it is considered with the selected behavior factor q. The philosophy of both factors is the same, i.e. to account for the capability of ductile members to deform beyond their yield point. However, there are two main differences between the q and the m-factors. The most important difference is that, whereas the m-factors are member specific (i.e. different m-factors may be assigned to the different structural components), the q-factor is based on the entire capability of the building to absorb energy. Secondly, the m-factors operate on the capacity side of the inequality, effectively increasing the strengths, whereas the q-factor is employed to decrease the demand on the components (both factors assume values equal or larger to unity).

Component Demands: For the deformation-controlled actions the component demand is calculated from the set of linear analyses. For the force-controlled actions the component demand is calculated based on capacity design considerations (i.e. estimate of the maximum action that can be developed in a component, based on a limit-state analysis), taking into account the expected strength of the components that deliver forces to the component under consideration. This is done in order to make sure that a failure in the force-controlled action is avoided. For example the shear demand on a column is not considered directly from the analytical calculations, but rather by the expected bending capacity of the member at its two edges, which is often very conservative.

Component Capacities: For the deformation-controlled actions the capacity of the components shall be based on the expected strengths. On the contrary, for the force-controlled actions the capacities shall be based on lower-bound strengths.

Component Demands: For the deformation-controlled actions the quantities checked are deformations (rather than forces), as these are calculated from the nonlinear analysis. For the force-controlled actions, the component demands are again forces, but now they are calculated directly from the nonlinear analysis. It is stressed that the demands are not to be determined from capacity design considerations as in the linear case, since inelasticity is explicitly accounted for by the nonlinear analysis method, and the calculations are significantly more accurate.

Component Capacities: For the deformation-controlled actions the component capacities are taken as permissible inelastic deformation limits at the target displacement. For the force-controlled actions, the component capacities are taken as lower-bound strengths at the target displacement. Contrary to the deformation-controlled actions, the checks for the force-controlled actions are performed again in terms of forces.

In order to investigate the effect that the method of analysis has on the outcome of the structural assessment and retrofit methodologies, two application examples will be examined. The first example refers to the assessment of an industrial building with short columns, whereas the second example is a real case study with the strengthening of a small residential building with a soft ground floor that has been severely damaged from two consecutive earthquakes. All the analyses and all the checks have been carried out with SeismoBuild, a package dedicated to the seismic assessment and strengthening of existing buildings, which is capable of performing linear and nonlinear, static and dynamic analysis. More information on the two examples can be found at [1].

In the current section the structural assessment and strengthening of an industrial building with short columns will be presented. The building is a typical design of the late 1980s and it consists of two rectangular floors of approximately 880m2 each (Figure 1).

Figure 1: 3D rendering of the industrial building

The concrete grade is C16/20, the steel grade is S400 for longitudinal bars and S220 for the stirrups. There is adequate longitudinal reinforcement (e.g., 8∅20 rebars in the columns), however the shear reinforcement is just Φ8/15 for the beams and Φ8/30 for the columns. The infilled walls form a series of vulnerable, short columns in the entire building perimeter. These short columns are the most important characteristic of the building and constitute a serious structural problem related to its seismic behavior.

The building will be analyzed according to ASCE 41 and ACI 369.1 for a single Performance Objective ‘g’ (see Table C2-8 in ASCE 41-23), which combines the (3-C)

Performance Level for Life Safety, and the BSE-1E Seismic Hazard Level with a 20% probability of exceedance in 50 years.

Code-based checks in shear and bending for all the members were carried out. Bending is a deformation-controlled (ductile) failure mechanism, for which reason the checks were performed in terms of forces (bending moments) for the linear analyses, but in terms of the plastic hinge rotation in the nonlinear methods. All the checks are expressed in terms of the demand-to-capacity ratio (DCR) for every member. The DCR is the proportion, by which the demand is larger than the capacity; if DCR<1 the member is safe, if instead DCR>1 the member fails.

The checks have been carried out for both shear and plastic hinge rotation for all the members. No failures have been observed for the checks in plastic hinge rotation, and the maximum DCR ratios was 0.536.

On the contrary several failures were observed in the shear checks. As expected, almost all the failures were located at the short columns in the perimeter, confirming the fact that lightly reinforced short columns are indeed an element of increased vulnerability in existing buildings. DCRs range from 1.236 to 1.431 in the ground floor, and from 1.040 to 1.274 in the second floor (Figure 2). In total 44 members have failed.

Figure 2: Shear checks with the Nonlinear Static Procedure, NSP

The 8 linear dynamic analyses required by ASCE-41 ran faster than the 8 pushover analyses (3 seconds, instead of 27 seconds in an average Intel i7 processor – both sets of analyses were executed using the advanced parallel computational capabilities of SeismoBuild). However, the shear demand with LDP on the short columns is considerably higher (up to 664 kN with LDP, with respect to a maximum of 285 kN with NSP). It is noted that shear is a brittle type of failure and the demand is calculated with capacity design considerations. taking into account the expected strength in bending of the components that deliver forces to the component. This is done, in order to make sure that a failure in the force-controlled action is avoided, since the results from the elastic analyses are expected to be quite inaccurate.

The higher demand is obviously reflected on the results of the capacity checks in shear. The maximum demand-to-capacity ratio in shear is now 5.17, and almost all structural members fail (Figure 3). In total 112 members have failed, all 72 columns and 40 of the beams.

Figure 3: Shear checks with the Linear Dynamic Procedure, LDP

Bending is a ductile action, and the checks in the linear methods are carried out in terms of bending moments, which is usually more conservative. This is highlighted in Figure 4 that shows the bending checks. Whereas with pushover analysis, no failures are observed (the maximum DCR is equal to 0.536), with the linear dynamic procedure 59 failures occur and the maximum DCR is equal to 2.678.

Figure 4: Bending checks with the Linear Dynamic Procedure, LDP

The Nonlinear Dynamic Procedure is generally more expensive in terms of computational resources (NDP: 7 minutes and 18 seconds for 11 dynamic analyses. NSP: 27 seconds for 8 pushover analyses). One additional complication with the NDP is the selection, scaling or matching to the target spectrum of the accelerograms, with which the dynamic analyses are run. Note however that in the case of SeismoBuild this process does not pose a significant challenge, since SeismoBuild utilizes several SeismoArtif algorithms for the automatic creation of artificial accelerograms that match the acceleration spectrum for the different seismic hazard levels.

These complications however are compensated with more accurate, and less conservative results. Now the maximum demand-to-capacity ratio is now 1.309, which is 9% less than the 1.431 value of the maximum DCR in the NSP. Moreover, only 20 members have failed, as opposed to the 44 members that failed with pushover analysis (Figure 5). The additional computational cost pays off with more accurate and less conservative results, and consequently lighter structural interventions for the seismic strengthening.

Regarding the bending checks, again no failures have been observed in plastic hinge rotation, as in the case of the nonlinear static procedure. However, the DCR ratios are now decreased; 0.326 in the NDP as opposed to 0.536 in the NSP.

Figure 5: Shear checks with the Nonlinear Dynamic Procedure, NDP

In this section the structural assessment and strengthening of a relatively small 2-storey residential building is presented. The building was constructed in the 1960s (first, ground floor) and the 1980s (second floor), and it had a soft ground floor with extremely weak columns, some of which were confined by infilled walls that form short columns. All the ground floor columns (18 in total) were damaged during the two 2014 Kefalonia earthquakes in Greece, of magnitudes 6.1 and 6.0, and four of the columns were very severely damaged with the complete deterioration of the concrete, the fracture of hoops and the local buckling of the longitudinal reinforcement (see [5] and Figure 6).

Figure 6: Damage sustained by the 2-story residential building

The concrete grade was found approximately equal to C16/20 and the steel grade was S220. The members were lightly reinforced, especially in shear (e.g., hoops Φ6/50 in the columns).

The building was analyzed according to the provisions of Eurocode 8, Part-3 for the Limit State of Significant Damage SD. The peak ground acceleration of the region is 0.36g and the analyses were carried out with a 10% probability of exceedance in 50 years, i.e. a return period of 475 years.

The checks have been carried out in shear and chord rotation for all the members. The values of the demand-to-capacity (DCR) ratios for the checks in shear and bending with the Nonlinear Static Procedure are displayed in Figure 7 and Figure 8, respectively. The maximum DCR ratios are 3.500 in shear and 2.803 for chord rotation. This is in accordance with the on-site, post-earthquake observations and indicates that the building would require strengthening even it was not damaged from the seismic events.

Figure 7: Shear checks with the Nonlinear Static Procedure, NSP

Figure 8: Chord rotation checks with the Nonlinear Static Procedure, NSP

It is noteworthy that the demand exceeds the capacity in all 18 columns of the ground level in shear, and in 11 out of the 18 ground floor columns in chord rotation, confirming again the increased vulnerability of soft ground stories and short columns in older, lightly reinforced buildings. What is also very interesting is the fact that the nonlinear static analysis of SeismoBuild managed to identify correctly the locations where increased damage occurred (for more information see [1]).

As expected, with the Nonlinear Dynamic Procedure the demand-to-capacity ratios are generally smaller (maximum DCR equal to 2.393 in shear and 0.659 in bending). What is more noteworthy however is that the DCR values fitted the locations and the extent of damage even better than the NSP (as explained in [1]), another indication that the NDP is the most accurate method for structural analysis.

On the contrary, with the Linear Dynamic Procedure the demand-to-capacity ratios are very high with values up to 6.720 in shear and 11.031 in bending. In total 43 out of the 80 structural elements have failed (Figure 9). Now the failures are not confined to the ground floor columns; in several beams of the first floor and columns of the second floor the demand exceeds the capacity. This differs from what has been observed from the post-earthquake survey, and it is yet another indication that the linear methods of analysis are not suitable for older, weak buildings.

Figure 9: Shear checks with the Linear Dynamic Procedure, LDP

Because all the columns at the ground level were severely damaged, the construction of strong reinforced concrete jackets was actually the only technically acceptable solution for the strengthening of the building. The retrofit scheme consisted of the strengthening of all the columns at the soft storey with a 10 cm wide reinforced concrete jackets; the other structural members were left unstrengthened.

With the Nonlinear Static Procedure NSP none of the member fails. The maximum DCR ratios are now observed in the unstrengthened beams with values up to 0.982. For the columns that have been jacketed the maximum DCR is equal to 0.568, considerably smaller than the 3.500 value of the original structure (Figure 10). The maximum DCR in the vertical members is found on the second floor, which was not strengthened (DCR=0.978). Similar are the observations for the checks in chord rotation with a maximum DCR value equal to 0.413.

Figure 10: Shear checks in the strengthened building with the Nonlinear Static Procedure, NSP

With the Nonlinear Dynamic Procedure NDP the results are better in terms of the maximum demand-to-capacity ratios; the maximum DCR in shear is equal to 0.881, and the maximum DCR in chord rotation is 0.201.

With the Linear Dynamic Procedure LDP instead the results are extremely conservative. Although the jacketed columns of the ground floor withstand the demand, in total 27 members fail in shear and 33 members fail in bending. The maximum DCR values are 3.345 in shear and 10.331 in bending (Figure 11).

Figure 11: Shear checks in the strengthened building with the Linear Dynamic Procedure, LDP

Obviously, if the design of the strengthening scheme was based on the results from the linear dynamic analysis, the interventions would have been much more costly and more invasive, since retrofit of the components in the second (undamaged after the earthquake) floor would have also been required.

The outcome of the current investigation shows that the nonlinear methods are much more suitable for analyzing existing buildings. This is because the liner methods, due to the inherent inaccuracies, are overly conservative in their estimates of the seismic capacity and very frequently they lead to excessive interventions that are costly and cause significant and unnecessary disturbance in the operation of the building.

The use of nonlinear methods instead, combined with the good knowledge of the structural configuration can be very advantageous, and lead to lighter and less invasive interventions. In particular, the NSP is gradually becoming the ‘standard’ methodology for assessment and retrofit, because it is faster than the NDP, but also because of the simplicity in its application.

[1] Antoniou, S. (2025). Linear vs. nonlinear procedures for the seismic assessment and retrofit of existing RC buildings. In Proceedings of the 10th International Conference on Computational Methods in Structural Dynamics and Earthquake Engineering (COMPDYN 2025) (Paper C 27254). Rhodes Island, Greece (Available at URL: https://2025.compdyn.org/proceedings/pdf/27254.pdf).

[2] Antoniou, S. Seismic Retrofit of Existing Reinforced Concrete Buildings, 1st Edition. John Wiley & Sons Ltd, 2023.

[3] F. Bianchi, M. Guidotti, R. Pinho, R. Nascimbene, Seismic assessment of RC buildings using linear analysis methods – an overly conservative practice. Proceedings of the 18th World Conference on Earthquake Engineering, WCEE 2024. 30 June – 5 July, 2024.

[4] Meral, E., Cayci, B.T. and Inel, M.. Comparative study on the linear and nonlinear dynamic analysis of typical RC buildings. Revista de la Construcción. Journal of Construction, 23(3), 587-607, 2024.

[5] Alfakat S.A. Repair and strengthening of a 2-storey residential building that was severely damaged during the 2014 Kefalonia earthquakes. Available at: https://www.alfakat.gr/en/project/repair-strengthening-building-severely-damaged-2014-kefalonia-earthquakes/ (Accessed: December 18, 2025).

[6] Structural Assessment, Strengthening & Retrofitting carried out using SeismoSoft Earthquake Engineering Software.

We are pleased to invite all structural engineers, researchers, and industry professionals to our upcoming free webinar, “SeismoBuild in Action.” This session is designed to give you a comprehensive walkthrough of SeismoBuild, our premier software solution dedicated to the assessment and strengthening of existing reinforced concrete structures.

Whether you are new to the software or looking to refine your existing workflow, this webinar will demonstrate how to go from a blank slate to a fully compliant retrofit report efficiently.

Event Details

• Date: Thursday, 18 December 2025
• Time: 16:00 CET
• Duration: 90 minutes (Presentation + Live Q&A)
• Format: Online Webinar
• Language: English
• Cost: Free of Charge

What You Will Learn

In this practical, hands-on session, our experts will guide you through the complete assessment lifecycle:

• Efficient Modeling: How to quickly set up and define a structural model in SeismoBuild.
• Defining Parameters: Best practices for assigning materials, structural members, and load cases.
• Code-Compliance: Performing rigorous code-based checks and accurately interpreting the analysis results.
• Automated Deliverables: How to automatically generate detailed CAD drawings and technical descriptions for your project.
• Live Interaction: A dedicated Q&A session to answer your specific technical questions.

How to Register

Reserve your seat today to ensure you don’t miss this opportunity to enhance your expertise in seismic retrofitting.

Can’t make it live? We understand that schedules can be busy. Please register anyway, and we will send a recording of the full session to your email after the event concludes.

We look forward to welcoming you online on December 18th!

We are thrilled to announce our participation and strong presence at the 6th Panhellenic Conference on Earthquake Engineering and Technical Seismology (6PSAMTS), taking place in Athens, Greece, on October 30-31 & November 1, 2025.

Co-organized by the Greek Association for Earthquake Engineering (Ε.Τ.Α.Μ.) and the Technical Chamber of Greece (TEE), this is the premier event for the seismic engineering community in Greece and Cyprus. It serves as a vital platform for discussing the latest advancements in seismic risk management, new Eurocodes, and innovative retrofitting technologies—areas where our software solutions are essential tools.

Join Us at Stand 7!

The SeismoSoft team will be showcasing our complete suite of software solutions for structural assessment, strengthening, and retrofitting at Stand 7 in the exhibition hall.

We invite all attendees—engineers, researchers, and students—to stop by our stand for a personalized demonstration of how our programs, including SeismoBuild and SeismoStruct, can streamline your non-linear analysis and help you achieve compliance with seismic codes.

We look forward to reconnecting with our long-standing users and engaging in meaningful discussions on advancing seismic resilience across the region.

We hope to see you in Athens!